Optica Applicata Vol X X V I , No. 1, 1996
Transversal and axial gains in the confocal scanning microscope
of leaky annular pupils
A. Magiera
Institute of Physics, Technical University of Wroclaw, Wybrzeże Wyspiańskiego 27, 5 0 — 370 Wroclaw, Poland.
The case of superresolution by positive transmitting pupil annular filters, reported in [1], has been extended to include the case of negative transmitting annular filters. The transversal and axial superresolution of confocal scanning microscope for the intensity spread function corresponding to several values o f the strength c is
considered-1. Introduction and the numerical results
In the paper [1], the transversal (Gr ) and axial (G J gains in confocal scanning microscope (CSM) with the leaky annular pupil P{p) — k were considered, where k is a positive amplitude transmittance of the filter (0 < k < 1). In the present paper, the case of transversal GT and axial GA gains in CSM for P(p) = k(p < e), where k is a negative amplitude transmittance of the filter ( —1 < fc < 0) is taken into account. The negative amplitude transmittance may be realized, for instance by introducing a delay plate of n phase shift into the central circle of the annular pupil filter.
The gain Gr is expressed by the formula [1]
and the axial gain GA is given by the formula [1]
GA = l . 2c(l — c) (1 + C )
The parameter c is expressed by k and ö in the form [1]
c Ö
k
(
1
)(2)
where: k is the amplitude transmittance in the annulus, Ô is the obstruction value of the annular pupil aperture.
In Figure 1, the runs of GT(c) and GA(c) are shown, while in Fig. 2 the case 0 < k < 1 considered by Sheppard [1] is presented.
The distribution of the normed transversal (U yyi^Jy)) point function in the focal plane is shown in Fig. 3, for the parameter values c = —0.1, —0.5, —0.7, —0.9. The distribution I(v) has been calculated numerically from the following dependence:
58 A. Magiera
Fig. 1. Axial GA and transversal GT gains in the parameter c for - 1 < k < 1
Fig. 2. Axial GA and transversal GT gains as a function of parameter c for 0 < k < 1. The case described in paper [1]. Enlarged fragment (A) of Fig. 1
Letter to the Editor 59
Fig. 3. Transversal /(v) intensity point spread function for parameter c = —0.1 (curve 1 \ c = —0.5 (curve 2), c — —0.7 (curve 3), and c = —0.9 (curve 4)
The normed axial distribution of the point spread function I{it)/ImajM) along the axis is shown in Fig. 4 for the parameter c = 0.5, —0.3, —0.9, —3.
For c = — 0.9 and u = 8.0 unnormed I(u) achieves the maximal I ^ u ) = 135.73; for c = -0 .3 and u = 4 I ^ u ) = 1.073; for c = - 3 and u = 8.8 I ^ u )
= 2.61; for c = 0.5 and u = 0 I maJu) = 1.
Fig. 4. Axial I(u) distribution of the intensity point spread function for parameter c equal successively to c = 0.5 (curve 1), —0.3 (curve 2), —0.9 (curve 3), and —3 (curve 4)
60 A. Magiera
The transversal distribution of the intensity spread function J(v) takes the following maximal values: I maJjv) = 21.17 for c = —0.9 and v = 3; I ^ v ) = 2.07 for
c = —0.7 and v = 2.4; I ^ v ) = 1 for c = —0.5 and v = 0; I^ J y ) = 1 for c = —0.1
and v = 0.
The superresolution resulting from the annular filtering according to the Sparrow criterion for the transversal intensity spread function J(v) and c = —0.9, —0.7 and for axial intensity spread function I(u) and c = —0.9, —3 has been shown in Figs. 3 and 4.
Reference
[1] Sheppard C. J. R., Optik 99 (1995), 32.
Received December 27, 1995 in revised form January 24, 1996
